Cellulose Biodegradation Models; An Example of Cooperative

Interactions in Structured Populations

Pierre-Emmanuel Jabin* Alexey Miroshnikov Robin Young

Abstract

We introduce various models for cellulose bio-degradation by micro-organisms. Those modelsrely on complex chemical mechanisms, involve the structure of the cellulose chains and areallowed to depend on the phenotypical traits of the population of micro-organisms. We then usethe corresponding models in the context of multiple-trait populations. This leads to classical,logistic type, reproduction rates limiting the growth of large populations but also, and moresurprisingly, limiting the growth of populations which are too small in a manner similar to theeffects seen in populations requiring cooperative interactions (or sexual reproduction). Thisstudy thus offers a striking example of how some mechanisms resembling cooperation can occurin structured biological populations, even in the absence of any actual cooperation.

Contents

1 Introduction 2

2 Cellulose bio-degradation: structure and mechanisms 52.1 Cellulose structure and enzyme systems . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Quantities to monitor and two basic bio-mechanisms . . . . . . . . . . . . . . . . . . 6

2.2.1 Cleaving Mechanism 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Cleaving Mechanism 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Cleaving Models 73.1 N -S-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Reduction to S-model for Cleaving Mechanism 1 . . . . . . . . . . . . . . . . . . . . 11

3.2.1 S-system with fast transitions of e2A and e2D . . . . . . . . . . . . . . . . . . 153.3 Cleaving Mechanism 2: T -model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Multiple trait T -model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4.1 Compatibility with the single trait model . . . . . . . . . . . . . . . . . . . . 19

4 Cooperation 204.1 Cooperation in the T -model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Asymptotics of B(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 Cooperation in multiple-trait T -model . . . . . . . . . . . . . . . . . . . . . . . . . . 224.4 Cooperative interactions in N -S-model . . . . . . . . . . . . . . . . . . . . . . . . . . 24

*Department of Mathematics, University of Maryland, College Park, pjabin@umd.eduDepartment of Mathematics, University of California, Los Angeles, amiroshn@gmail.comDepartment of Mathematics and Statistics, University of Massachusetts Amherst, young@math.umass.edu

1

5 A Model in the Continuous Setting 28

6 Numerical experiments 29

7 Appendix 317.1 Tail issue in deterministic selection dynamics . . . . . . . . . . . . . . . . . . . . . . 31

1 Introduction

The goal of this article is to derive models for structured populations of micro-organisms living offcellulose degradation. Our first step is to study the mechanisms by which some micro-organismscan use cellulose. The full process is obviously complex and we have to abstract its most importantfeatures. This gives us a hierarchy of models, depending on the level of simplification that onedesires.

The second step is to couple those models with the population dynamics of the correspondingmicro-organisms. While the mechanism of bio-degradation that we consider is similar for eachspecies of micro-organisms, we allow for some variability from one species to another, in the enzymesinvolved for instance. This leads to a population structured by a phenomenological trait thatdescribes the exact path of bio-degradation.

As the amount of cellulose is limited, the total growth of the population is, unsurprisingly,limited as well. More interesting are the effects when the total population or the population in agiven species is small. The model does not include any actual cooperation between micro-organismsbut as the bio-degradation occurs in several steps, the process is nonlinear in the population sizeeven if cellulose is abundant. This puts small populations at a disadvantage, introducing an effectsimilar to classical cooperation.

Cellulose Bio-degradation. Mechanisms and Models. Cellulose is the structural componentof many plants and is therefore the most abundantly produced bio-polymer; it is a homo-polymerconsisting of a vast number of glucose units. The most important feature of cellulose as a substrateis its insolubility. As such, bacterial and fungal degradation of cellulose, (e.g. by fungi Trichodermareesei or bacteria Clostridium thermocellum), occurs exocellularly. The products of cellulose hy-drolysis are available as carbon and energy sources for microbes that inhabit environments in whichcellulose is biodegraded [31, 32].

In this work we model cellulose bio-degradation as a multiple-step process, reflecting realisticmechanisms described in [32]. Let %(t) denote the mass of cellulose. The biodegrading microorgan-ism is unable to consume (degrade) the cellulose % directly. Instead, the individuals produce twoenzyme complexes e1(t) and e2(t) that act in a two-stage process.

During the first stage, the (endoglucanase) enzyme e1 weakens cellulose fibers in %: that is,it randomly cuts the fibers, creating the so-called reducing and non-reducing ends which serve aslanding sites for the (exoglucanase) enzyme e2. During the second stage, the enzyme e2 locates alanding site and attaches itself to it. Once attached, it cleaves off cellobiose (a major energy sourcefor the microorganisms) from the chain of polysaccharides. Some portion θp ∈ [0, 1] of cellobiose isconsumed directly by the microorganism that produced the enzymes, and the rest is available forother individual microorganisms in the population due to diffusion. The above mechanisms can beviewed as follows:

Growth of micro-organisms + influx of cellulose %(t)

2

↓Production of enzyme complexes e1(t) and e2(t)

↓Weakening of %(t) by e1(t)

↓Production of cellobiose p(t) by e2(t) acting on %(t).

The last two steps in the above diagram constitute the so-called cleaving mechanism. In ourwork we present two different cleaving mechanisms that differ in complexity (see Section 2). InCleaving Mechanism 1 the enzyme e2 that cuts off cellobiose units has two states, ‘attached to’ and‘detached from’ cellulose. In Cleaving Mechanism 2, however, the enzyme e2 is always detached. Inthis mechanism cleaving happens instantaneously once the enzyme e2 finds a spot on the cellulosewhere it is able to cut off cellobiose.

In the present work we develop several models of varying complexity which incorporate thesemechanisms. Even in the simplest model the aforementioned cascade of events produces a coopera-tive effect, which appears due to the fact that the cellobiose units cleaved off by the enzyme of onemicroorganism are available for consumption by other individuals located nearby. Mathematically,these effects are encoded in the reproduction rate B(n) of the population n. In particular, for smallpopulations, the population size n(t) turns out to behave as

∂tn(·, t) ∼ n(B(n)− d

)with B(n) ∼ Cn2 when n� n̄,

where n̄ is a critical threshold.In general, the population includes various species of micro-organisms. In that case, the exact

enzyme complexes used may change across species. We represent the different species xj by traitsj ∈ {1, . . . ,M}, where the population with trait j uses enzymes complexes denoted e1,j and e2,j .This can be included in a more general framework by considering continuous traits x with sub-populations n(t, x) with a model of the form

∂tn(x, t) =(B[n](x, t)− d(x)

)n(x, t) , (1.1)

where B[n] is now an integral operator; see (5.2) for the precise formula. The case of discretetraits is included in this framework by taking n(t, x) =

∑j nj(t) δxj (x) with nj(t) the population

of individuals with trait xj .

Hierarchy of Models. Let us give a brief description of the models developed in our work andtheir relations.

Mechanism 1︷ ︸︸ ︷(N -S-model)→ (S-model)→

Mechanism 2︷ ︸︸ ︷(T -model)← (multiple-trait T -model) .

The most complex model that explicitly takes the kinetics of enzymatic reactions into accountis the N -S-model. We monitor the evolution of the cellulose chains N , structured by polymerlength and the total number of landing sites where the enzyme e2 can be attached. This modelincorporates cleaving Mechanism 1, in which the enzyme e2 is either ‘attached’ (e2A) or ‘detached’(e2D). In addition, the model tracks the evolution of unoccupied landing sites S and the totalnumber of landing sites T ; these two variables are related via T = S + e2A. Under the assumptionthat the cleaving rates are independent of the polymer structure, the N -S-model reduces to the

3

S-model. The next reduction occurs when cleaving Mechanism 1 is replaced with Mechanism 2 inwhich the two stages of cleaving are combined into one. Here, the enzyme e2 is always detached(the attachment and cleaving of cellobiose occur instantaneously) while the number of unoccupiedlanding sites and total number of sites coincide T = S, leading to the T -model. Though this is thesimplest model it still captures the basic features of cellulose biodegradation. For this reason weextend it to a multiple-trait T -model which allows for species structured by a parameter. Finallyto study more specifically cooperative effects we modify the above models to take only time scaleson which the population changes into account (see Section 4).

This framework has some interest for the analysis even when it is only applied for a finitenumber of traits as is the case here. Indeed our traits correspond to possible enzyme complexesand while some of them may still be unknown, we do not expect their number to be very large. Wealso refer to [18] for the reason that in general one can only expect a finite (if possibly very large)number of traits at the ecological equilibrium, Evolutionarily Stable Strategy or ESS.

Our model therefore resembles systems of population dynamics, see [25] for instance. However,in contrast to many of those systems, the cellulose bio-degradation process leads to both competitionbetween individuals and species (for the resource) and cooperative interactions. This occurs at theinterspecies level as the byproduct of the process, cellobiose, is the same independent of the enzymecomplexes involved, and can benefit any individual in the population (and not only individuals usingthe same complexes). Cooperation also occurs specifically within each species (or between speciesthat are close enough). This follows from the fact that an individual with similar enough enzymecomplexes can use a landing site created in the cellulose by the endoglucanase enzyme complex ofanother individual. The mathematical models developed in our work thus lead to different (andhopefully improved) phenomenological results for small populations (in deterministic models) sincecooperation significantly affects the dynamics, as discussed in more detail below. Those differencesof behaviour for small populations may not impact the final ESS, but they are important in thetransitory regime, in particular in the presence of mutations.

From the ecological point of view, an important conclusion of our modeling is that as soon asminimally complex biochemical processes are involved, one cannot simply interpret the relationbetween the individuals in the population (or in different sub-populations) as competitive or co-operative. When the population is very large, the interaction looks competitive because resourcesare limited. On the other hand when the population is very small, the interaction seems to becooperative as the limitation on growth mostly comes from the ability of the individuals to processefficiently the several steps of the biochemical process. But this is only a caricature of the actualinteraction which cannot be reduced to pure cooperation or competition.

Tail issue in deterministic selection dynamics. The cooperative nature of the interactionwhen populations are small also becomes important for deterministic selection models. As sub-populations may grow or decrease exponentially, there typically are several orders of magnitudebetween the large populations of dominant traits and smaller ones. This poses an acute problemfor modeling as one would need to use deterministic model for the larger populations. But suchdeterministic equations do not adequately capture the stochastic nature of the dynamics of smallerpopulations. This problem can however be alleviated if such small populations go extinct sinceaccurately modeling their behavior loses its relevance. This is precisely what happens if cooperationis needed when the population goes below a certain threshold: The birth rate of a small populationis then necessarily too small ensuring a negative reproduction rate and extinction. We furtherdiscuss this phenomenon in the appendix.

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Structure of the paper. In Section 2, we set notation and, following the seminal article [32],introduce basic processes and mechanisms that constitute cellulose biodegradation. In Section 3,we develop a series of mathematical models, that differ in complexity, for the coevolution betweenmicroorganisms which consume cellulose and the cellulose chains. In Section 4, we carry outa qualitative analysis of the models developed in Section 3 under the assumption that cellulosedynamics and enzymatic reactions occur on a faster time scale than the dynamics of the microbialpopulation. In Section 5, the multiple-trait model, developed earlier in Section 3, is extended toa continuous-trait model. We carry out some numerical experiments in Section 6, in which wecompare the models and demonstrate that the T -model can be regarded as a limit of the S-model.Finally, we discuss the tail issue in more detail in the appendix.

2 Cellulose bio-degradation: structure and mechanisms

2.1 Cellulose structure and enzyme systems

Cellulose is the most abundantly produced bio-polymer. It is a homo-polymer consisting of glucoseunits joined by β-1, 4 bonds. In secondary walls of plants, the size of cellulose molecules (degreeof polymerization) varies from seven thousand to fourteen thousand glucose moieties per molecule.Cellulose molecules are strongly associated through inter- and intra-molecular hydrogen-bindingand van der Waals forces that result in the formation of microfibrils, which in turn form fibrils.Cellulose molecules are oriented in parallel, with reducing ends of adjacent glucan chains located atthe same end of a micro-fibril. These molecules form highly ordered crystalline domains interspersedwith more disordered, amorphous regions. Although cellulose forms a distinct crystalline structure,cellulose fibers in nature are not purely crystalline. The degree of crystallinity varies from purelycrystalline to purely amorphous.

To degrade plant cell material, microorganisms produce multiple enzymes known as enzymesystems [32]. For microorganisms to hydrolyze and metabolize insoluble cellulose, extra-cellularcellulases (degradation enzymes) must be produced that are either free or cell associated. Microor-ganisms have adapted different approaches to effectively hydrolyze cellulose, naturally occurringin insoluble particles. Cellulosic filamentous fungi (and some types of aerobic bacteria) have theability to penetrate cellulosic substrates through hyphal extensions, thus often presenting theirfree cellulase systems in confined cavities within cellulosic particles. In contrast, anaerobic bacte-ria lack the ability to effectively penetrate cellulosic material and perhaps had to find alternativemechanisms for degrading cellulose. This led to the development of complexed cellulase systems(called cellulosomes) which position cellulase producing cells at the site of hydrolysis, as observedfor clostridia and ruminal bacteria.

Overall there are three major components of cellulase systems: (i) endoglucanases, which ran-domly hydrolyze β-1, 4 bonds within cellulose molecules, thereby producing reducing and non-reducing ends; (ii) exoglucanases, which liberate (cleave off) either glucose (glucanohydrolases) orcellobiose (cellobiohydrolase) that serve as major products from reducing or non-reducing ends ofcellulose polysaccharide chains; and (iii) β−glucosidases which hydrolyze cellobiose yielding glucose(the major product of cellulose hydrolysis used by microorganisms as energy source). For detailssee [32].

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2.2 Quantities to monitor and two basic bio-mechanisms

We first consider the case of populations with one trait. In our analysis we let n = n(t) denote thetotal number of the microorganism that degrades cellulose, while the total number of endoglucanasesand exoglucanases produced by the microorganism are denoted by e1 = e1(t) and e2 = e2(t),respectively.

We view cellulose as a crystalline conglomerate of fibers (chains of polysaccharides). Accordingto our discussion above, during the first stage of the degradation the endoglucanase enzyme e1weakens fibers, which means that e1 randomly cuts the fibers by creating reducing and non-reducingends that then serve as landing sites for exoglucanases e2. Viewing cellulose as a three-dimensionalstructure, one can imagine that structure with punctures or cuts after the first stage. It is still thesame cellulose but with more ‘cuts’ that serve as landing sites for the exoglucanase enzymes e2.

There are two different complexes of exoglucanases that are able to cleave off (liberate) cellobioseunits from the cellulose chains. These are exoglucanase CBHI and exoglucanase CBHII. Theenzymes of the first type are able to attach to reducing ends, and the second to non-reducingends (see [32, p. 512]). In our models, for simplicity, we do not distinguish between the twotypes; both are represented by e2. For this reason, we do not differentiate between reducing andnon-reducing ends, and call them ‘landing sites’ instead. We add that each landing site (createdby the endoglucanase e1) always contains one reducing end and one non-reducing end. In ourmodel, however, we allow each landing site to host only one enzyme. Once the exoglucanase e2lands on the chain, it cleaves off cellobiose from the chain of polysaccharides. In this treatmentwe do not consider the third type of enzyme (β−glucosidase) and treat cellobiose as the majorproduct of degradation. We instead assume that some portion θp ∈ [0, 1] of cellobiose is consumedby the microorganism that produced it, while the rest diffuses and is freely available for generalconsumption. We let p(t) denote the total of the freely available cellobiose.

Thus in our model, there are two main stages in which the exoglucanase e2 produces cellobiosep(t): first, the enzyme locates a landing site and attaches itself to the chain there; next, it keepscleaving off cellobiose units at a certain rate until it either disintegrates or detaches from the chain.This leads to two basic modeling approaches, which we call cleaving mechanisms.

2.2.1 Cleaving Mechanism 1

Since the time spent by an individual exoglucanase enzyme e2 locating a landing site may differsignificantly from the time it is attached to the landing site, it is useful to consider two states forexoglucanase, namely detached and attached states. In the first mechanism we distinguish them,letting e2D(t) represent the amount of detached e2 (which may wander freely or on a leash, that isattached to a bacterial cell wall), and e2A(t) the amount attached to a landing site. Denoting thetotal number of landing sites by T (t) and the number of unoccupied landing sites by S(t), it thenfollows that

T (t) = S(t) + e2A(t) . (2.1)

We suppose that, at any moment of time, unoccupied spots S(t) become occupied (or attacked)by the detached enzymes e2D at a certain rate to be described. Next, we assume that an individualattached enzyme e2A cleaves off cellobiose units from the cellulose chain, again with a given rate.Also, we assume that some proportion of the (attached) enzyme e2A detaches from a landing site,and that some fraction θr ∈ [0, 1] of those sites become unavailable for landing, that is, the landingsites are destroyed.

6

2.2.2 Cleaving Mechanism 2

The second mechanism is somewhat simplified. It may be used to describe complex cellulases whereexoglucanases are not entirely free (they are attached to bacterial cell walls, and once bacteria leavesthe spot the enzyme becomes detached from the landing site as well). Here we suppose that at anymoment of time, all existing T (t) landing sites are available for an attack by the enzyme e2. Thelanding sites T (t) are attacked with a certain rate b(T (t)) and a certain (average) amount q > 0of cellobiose units is cleaved off by each individual enzyme e2, after which the enzyme e2 detachesitself. We view such an attack as instantaneous. Thus, after such an (instantaneous) attack, allT (t) landing sites are again unoccupied. We will assume that after the attack a certain portionθr ∈ [0, 1] of the (attacked) landing sites become unavailable, that is, destroyed. In that scenariothe two processes, finding a landing site and cleaving off cellobiose, are lumped together (with ahidden assumption that enzymes cannot remain attached to a chain for a very long time).

Remark 2.1. This first mechanism is more realistic since it takes into account time spent by theenzyme on a site. This mechanism can be employed for modeling systems where both non-complexand complex cellulases are present.

Remark 2.2. In the second mechanism the landing sites T (t) serve as ”prey” and e2 as ”predators”,with one difference: the enzymes attack the prey, use it and leave it alone. After an attack only acertain proportion of the sites is destroyed, while the rest is still usable.

3 Cleaving Models

We now develop several models of varying complexity to describe cellulose biodegradation. Webegin with a one-trait model describing cleaving mechanism 1, which retains the most detailedcellulose structure. This N -S system is too cumbersome to be effectively analyzed, so we reduceit by removing the detailed cellulose structure, to obtain the S system, a set of 7 equations whichmodels cleaving mechanism 1. We then modify the S system to model cleaving mechanism 2, whichyields the T system, consisting of 6 equations. Finally, we adapt the simplest of these, the T system,to a multiple trait model, in which there can be several species of microorganism consuming thesame cellulose.

3.1 N-S-model

We first consider a model in which we monitor groups of cellulose chains consisting of l ≥ 1 cellobioseunits; in that case we say that the chain has length l. This allows us to develop a fundamentalmodel incorporating cleaving mechanism 1.Assumptions and notation. Cellulose chains may have different topological configurations: theycould be linear or rectangular (when fibers are embedded in a lignin matrix) or they could havea random three-dimensional structure. Monitoring the topology increases the complexity, but itdoes not provide a better tool for studying the population dynamics. After all, it is the number oflanding spots that matters rather than the configuration of the cellulose chains. Thus we make noassumption about the configuration and monitor only the length of its constituent pieces. Anotherassumption we make is that the enzyme e1 produces a landing site without physically cutting thechain. This assumption decreases complexity in the model while it does not change the dynamics.Indeed, if we allow e1 to physically cut a chain in the model, then the chain could be split into two

7

parts when e1 acts. In that case, the number of landing sites would be the same as in the scenariowhen the landing site is created without a physical cut. Finally, we impose the requirement thatat most one landing site per unit of cellobiose is allowed; this reflects the fact that cellulose chainsrepresent systems of discrete units.

We say that a cellulose chain is in the (l, i)-state, or is an (l, i)-chain, if it has length l (soconsists of l ≥ 1 cellobiose units) and i ∈ {0, 1, . . . , l} landing sites (previously made by enzymee1), and let N

l,i(t) denote the number of (l, i)-chains. Recall that p denotes the number of cellobioseunits available for general consumption, e1 denotes the number of endoglucanase enzymes, whichproduce landing sites, and e2D denotes the number of detached exoglucanase enzymes. We refinethe attached exoglucanase to refer to those enzymes attached to chains in the (l, i)-state by el,i2A,where the superscripted indices (l, i) are in the set

(l, i) ∈ IL := {(l̃, ĩ) ∈ Z× Z : 1 ≤ l̃ ≤ L, 0 ≤ ĩ ≤ l̃} . (3.1)

Here L stands for the maximal number of cellobiose units in cellulose chains. We also use theconvention that

N l,i ≡ 0 if (l, i) 6∈ IL,

el,i2A ≡ 0 if (l, i) 6∈ IL or i = 0 .(3.2)

Enzyme dynamics. We assume that the rates of production of the enzymes e1, e2 by the microor-ganism and their degradation rates are fixed. The enzymes e1 and e2 are catalyzers which stay inthe system as long as they “live”. Then e1 satisfies

∂te1(t) = b1n(t)− d1e1(t) with b1, d1 > 0 . (3.3)

Next, the number of landing sites on (l, i)-chains is T l,i = iN l,i, so the number of unoccupiedlanding sites Sl,i(t) is

Sl,i(t) = T l,i(t)− el,i2A(t) = iNl,i(t)− el,i2A(t). (3.4)

Neglecting saturation effects, we suppose that unoccupied sites Sl,i are attacked by e2D with therate βl,iSl,i. Also, we assume that enzyme e2 located on a chain in (l, i)-state (randomly) detaches

from the chain with rate σl,i > 0. We let γl,ir > 0 denote the decay rate of an individual landingsite (whether it is occupied or not) and assume that if an occupied landing site degrades then theattached enzyme e2A disintegrates together with it. This leads to the following set of equationsthat monitor the dynamics of the enzyme e2:

∂te2D(t) = b2n(t)−∑l,i

βl,iSl,i(t) e2D(t) +∑l,i

σl,iel,i2A(t)− d2D e2D(t)

= b2n(t)−∑l,i

[βl,i(iN l,i(t)− el,i2A(t)

)e2D(t)− σl,iel,i2A(t)

]− d2D e2D(t)

∂tel,i2A(t) = β

l,iSl,i(t) e2D(t)−(σl,i + d l,i2A + γ

l,ir

)el,i2A

= βl,i(iN l,i(t)− el,i2A(t)

)e2D(t)−

(σl,i + d l,i2A + γ

l,ir

)el,i2A

(3.5)

where d2D > 0 and dl,i2A > 0 are the degradation rates of e2D and e

l,i2A, respectively, and b2 > 0 is

the production rate of e2, which equals that of e2D.

8

Chain dynamics. We neglect saturation effects and assume that the rate with which enzymes e1produce landing sites on chains in the (l, i)-state is

αl,i(l − i)N l,i(t)e1(t) = α̂l,iN l,i(t)e1(t) , with α̂l,i = αl,i(l − i) (3.6)

where the multiplier (l − i) reflects the requirement that only one landing site per cellobiose unitis allowed. We assume that a freshly made landing site cannot be instantaneously occupied anddifferent cuts don’t occur simultaneously, so that (3.6) is the transition rate of states (l, i)→ (l, i+1)due to the action of e1.

Recall that γl,ir > 0 is the decay rate of an individual landing site on a chain in (l, i)-state, sothat the rate of transition (l, i)→ (l, i− 1) due to degradation of landing sites is

γ̂l,iN l,i(t), with γ̂l,i = iγl,ir . (3.7)

Let ql,i > 0 denote the rate of production of cellobiose by an individual enzyme attached to an(l, i)-chain. Then, the total rate of cellobiose production by the enzymes el,i2A is

ql,iel,i2A(t). (3.8)

The rate ql,i can be expressed asql,i = cl,i + pl,i

where, cl,i is the rate of cleaving that results in the transition (l, i)→ (l − 1, i) and pl,i is the rateof cleaving that results in the transition (l, i) → (l − 1, i − 1). The latter transition occurs whenan enzyme el,i2A cleaves off a cellobiose unit and when moving along the chain reaches the nextcellobiose unit that also contains a landing site (this results in the decrease of landing sites by oneon the given chain). We note that cl,l = cl,0 = 0 and pl,0 = pl,1 = 0 for any l.

Let θr > 0 denote the proportion of landing sites that get destroyed after el,i2A detaches from a

chain or dies. This contributes to the transition (l, i)→ (l, i− 1) and the corresponding rate is

θ̂l,i el,i2A(t) with θ̂l,i = θr(σ

l,i + d l,i2A) (3.9)

Combining (3.6)–(3.9) we obtain the equations that monitor the dynamics of N l,i(t), namely

∂tNl,i(t) = rl,i +

(α̂l,i−1N l,i−1(t)− α̂l,iN l,i(t)

)e1(t) +

(γ̂l,i+1N l,i+1(t)− γ̂l,iN l,i(t)

)+(cl+1,iel+1,i2A + p

l+1,i+1el+1,i+12A − (cl,i + pl,i)el,i2A

)+(θ̂l,i+1el,i+12A (t)− θ̂

l,iel,i2A(t))− γl,i% N l,i(t),

(3.10)

where rl,i is the unit rate of production of cellulose, and γl,i% is the rate at which the cellulose natu-rally decays or becomes unavailable to the microorganism (that is, decay not directly attributableto the bacteria). We assume for simplicity that the cellulose provided by the environment has nolanding sites, so that rl,i = 0 for i ≥ 1, and in particular, the sum

∑i i r

l,i = 0.When polymer chains are long, that is l is large, and the landing sites are spaced out, which

happens when i is small relative to l, the coefficients pl,i can be neglected. To simplify the equation(3.10) we drop the coefficients pl,i except when i = l, corresponding to the boundary case when thenumber of sites l equals to the number of cellobiose units. The coefficients pl,l cannot be dropped

9

because cleaving cellobiose on the chain in state (l, l) always leads to (l − 1, l − 1). Moreover,dropping the coefficients pl,l would lead to a loss of conservation of the total cellulose in the systemwhen consumers are not present. This assumption leads to

ql,i = cl,i for l 6= i and ql,l = pl,l

and hence (3.10) becomes

∂tNl,i(t) = rl,i +

(α̂l,i−1N l,i−1(t)− α̂l,iN l,i(t)

)e1(t) +

(γ̂l,i+1N l,i+1(t)− γ̂l,iN l,i(t)

)+(ql+1,iel+1,i2A (t) + δliq

l+1,i+1el+1,i+12A − ql,iel,i2A(t)

)+(θ̂l,i+1el,i+12A (t)− θ̂

l,iel,i2A(t))− γl,i% N l,i(t).

(3.11)

where δli denotes the Kronecker delta.

Population dynamics. Let θp ∈ [0, 1] denote the proportion of produced cellobiose that becomesavailable for everyone. Then, using (3.8), the equations for the total amount p(t) of cellobioseavailable for everyone, and the total population n(t) of the microorganism are respectively

∂tp(t) = θp∑l,i

ql,iel,i2A(t)− γ n(t) p(t)− γp p(t), and

∂tn(t) =µn(t)

n̄+ n(t)

(γ p(t) + (1− θp)

∑l,i

ql,i el,i2A(t)

)− γn n(t),

(3.12)

where γ is the consumption rate, µ is the conversion efficiency, and γp, γn are decay rates of pand n, respectively. Here n̄ represents a critical population threshold: if n is large, n = O(n̄), thegrowth depends only on the cellobiose supply, while if n is small, n � n̄, the growth is linear butwith small growth rate, so the population is unlikely to survive (since µ/n̄ < γn).

Summary. Figure 1 shows possible states of the resource, state transitions and the rates at whichthese occur.

N l,iN l,i−1 N l,i+1

N l−1,iN l−1,i−1

N l+1,i N l+1,i+1

δliql,iel,i2A (1− δli)q

l,iel,i2A

α̂l,iN l,ie1θ̂l,iel,i2A + γ̂

l,iN l,i

α̂l,i−1N l,i−1e1 θ̂l,i+1el,i+12A + γ̂l,i+1N l,i+1

ql+1,iel+1,i2A δliql+1,i+1el+1,i+12A

Figure 1: Transition of cellulose chains.

10

Remark 3.1. The processes of creating a landing site or cleaving off a cellobiose unit may dependon the configuration of the chain, the crystallinity of the cellulose as well as the lifetime of theenzymes. Thus it is possible that within a given period of time (no matter how short) more thanone landing site is created or two or more cellobiose units are cleaved off from the same chain. Forsimplicity, we discount transitions other than (l, i) → (l, i + 1) and (l, i) → (l − 1, i), essentiallyassuming instantaneous transition. This approach is justified provided that the number and sizeof chains is very large compared to the amount of enzymes e1, e2, and the likelihood that twolanding sites are created or more than two cellobiose units are cleaved off from the same chainsimultaneously (or during a short period of time) is extremely small.

Another way to justify this assumption is to consider a time-continuous Poisson counting pro-cess, that corresponds to the events of creating a landing site and/or cleaving off cellobiose. Anyinstance when a landing site is created (that is the moment it becomes available for use by e2)or cellobiose unit is cleaved off the chain can be counted as an event. It is well-known that theprobability of two or more events happening instantaneously is zero (in other words the probabilitythat two events take place over the time ∆t is o(∆t); for details see [33, 40].

3.2 Reduction to S-model for Cleaving Mechanism 1

We now develop a simpler model by reducing the N -S-model, consisting of (3.3), (3.5), (3.11) and(3.12). It is convenient to allow the indices l and i to run through all of Z. This augments thepreviously defined system, but by choosing appropriate constants, we can ensure that N l,i and el,i2Avanish for all times whenever l ≤ 0, i < 0, or i > l and, in addition, el,02A = 0 for any l. We thendefine the quantities

%(t) =∑l,i

lN l,i(t), e2A(t) =∑l,i

el,i2A(t),

T (t) =∑l,i

T l,i(t) =∑l,i

iN l,i(t), S(t) =∑l,i

Sl,i(t),(3.13)

which represent the total number of cellulose units, attached exoglucanase enzymes, landing sitesand unoccupied sites, respectively. Note that each of these sums is finite provided we specifyappropriate initial conditions. We next assume that the constants are independent of l and i, sothat for each l, i ∈ Z,

βl,i = β , σl,i = σ , γl,ir = γr , dl,i2A = d2A, α

l,i = α, ql,i = q , γl,i% = γ% . (3.14)

Summing over l, i in (3.5) and using (3.4), we get

∂te2D(t) = b2n(t)− βS(t)e2D(t) + σe2A(t)− d2D e2D(t),∂te2A(t) = βS(t)e2D(t)−

(σ + d2A + γr

)e2A(t) .

(3.15)

To obtain equations for % and T , we scale and add equations (3.11). First, we recall∑l,i

i rl,i = 0 and define r =∑l,i

l rl,i =∑l

l rl,0 , (3.16)

11

so that r represents the total production of cellulose by the environment. Using (3.6) and makingthe change of variable j = i− 1 we obtain∑

l,i

i(α̂l,i−1N l,i−1(t)− α̂l,iN l,i(t)

)= α

∑l,i

(i(l − (i− 1))N l,i−1(t)− i(l − i)N l,i(t)

)= α

(∑l,j

(j + 1)(l − j)N l,j(t)−∑l,i

i(l − i)N l,i(t))

= α(∑

l,j

(l − j)N l,j(t))

= α(%(t)− T (t)

)and similarly, using (3.7) and j = i+ 1,∑

l,i

i(γ̂l,i+1N l,i+1(t)− γ̂l,iN l,i(t)

)= γr

∑l,i

(i(i+ 1)N l,i+1(t)− i2N l,i(t)

)= γr

(∑l,j

(j − 1)jN l,j(t)−∑l,i

i2N l,i(t))

= −γr∑l,j

jN l,j(t) = −γrT (t) .

Next, recalling that eL+1,i2A ≡ 0 for all i, we compute

L∑l=1

l∑i=0

i(ql+1,iel+1,i2A + δliq

l+1,i+1el+1,i+12A − ql,iel,i2A

)= q

L∑k=2

k−1∑i=0

iek,i2A − qL∑l=1

l∑i=0

iel,i2A + qL−1∑l=1

lel+1,l+12A

= −qL∑l=1

lel,l2A + qL∑l=2

(l − 1)el,l2A

= −qL∑l=1

el,l2A.

(3.17)

12

and, using (3.9), ∑l,i

i(θ̂l,i+1el,i+12A (t)− θ̂

l,iel,i2A(t))

= θr(σ + d2A)∑l,i

(i el,i+12A (t)− i e

l,i2A(t)

)= θr(σ + d2A)

(∑l,j

(j − 1) el,j2A(t)−∑l,i

i el,i2A(t))

= −θr(σ + d2A)∑l,j

el,j2A(t) = −θr(σ + d2A)e2A(t) .

Combining the above identities with (3.11) and (3.13) we conclude

∂tT (t) = ∂t

(∑l,i

iN l,i(t))

= α(%(t)− T (t)

)e1(t)− θr(σ + d2A) e2A(t)− (γr + γ%)T (t)− q

L∑l=1

el,l2A.

(3.18)

Next, referring to (3.4), subtracting ∂te2A from (3.18) and using (3.15), we obtain

∂tS(t) = α(%(t)− S(t)− e2A(t)

)e1(t)− q

L∑l=1

el,l2A − βS(t)e2D(t)

+(

(1− θr)(σ + d2A)− γ%)e2A(t)− (γr + γ%)S(t) .

(3.19)

We now multiply each term on the right-hand side of (3.11) by l and sum to get an equationfor %(t). First, using (3.6) and (3.7), we get∑

l,i

l(α̂l,i−1N l,i−1(t)− α̂l,iN l,i(t)

)= α

(∑i,j

(l − j)N l,j(t)−∑l,i

(l − i)N l,i(t))

= 0 ,

∑l,i

l(γ̂l,i+1N l,i+1(t)− γ̂l,iN l,i(t)

)= γr

(∑l,j

l j N l,j(t)−∑l,i

l iN l,i(t))

= 0 .

Similarly, we compute

L∑l=1

l∑i=0

l(ql+1,iel+1,i2A + δliq

l+1,i+1el+1,i+12A − ql,iel,i2A

)= q

L∑l=2

l−1∑i=0

(l − 1)el,i2A − qL∑l=1

l∑i=0

lel,i2A + qL−1∑l=1

lel+1,l+12A

= qL∑l=2

l−1∑i=0

(l − 1)el,i2A − qL∑l=1

l∑i=0

(l − 1)el,i2A − qe2A + qL∑l=2

(l − 1)el,l2A

= −qe2A.

(3.20)

13

and, using (3.9),∑l,i

l(θ̂l,i+1el,i+12A (t)− θ̂

l,iel,i2A(t))

= θr(σ + d2A)∑l

l[∑

i

el,i+12A (t)−∑i

el,i2A(t)]

= 0.

Combining the above expressions and using (3.13),(3.14) and (3.16), we obtain

∂t%(t) = ∂t

(∑l,i

lN l,i(t))

= r − q e2A(t)− γ% %(t) . (3.21)

Finally, by (3.8), (3.13) and (3.14), equations (3.12) become

∂tp(t) = θp q e2A(t)− γ n(t) p(t)− γp p(t)

∂tn(t) =µn(t)

n̄+ n(t)

(γ p(t) + (1− θp) q e2A(t)

)− γn n(t).

(3.22)

S-system. Note that the equation for S in (3.19) can be expressed as∂tS(t) = α

(%(t)− S(t)− e2A(t)

(1 +

qeb2Aαe1e2A

))e1(t)− β S(t) e2D(t)

+(

(1− θr) (σ + d2A)− γ%)e2A(t)− (γr + γ%)S(t) , eb2A :=

L∑l=1

el,l2A.

(3.23)

When the polymer chains are large, that is L >> 1, one would expect that the proportion ofcellobiose units lN l,l would be small compared to the total number of cellobiose % =

∑l,iN

l,i and

therefore one can expect e2Ae1 >> eb2A. Then, combining the above equations and dropping the

termq eb2Aαe1e2A

in (3.23) we obtain the S-system,Combining the above equations we obtain the S-system,

∂te1(t) = b1 n(t)− d1 e1(t)∂te2D(t) = b2 n(t)− β S(t) e2D(t) + σ e2A(t)− d2D e2D(t)∂te2A(t) = βS(t) e2D(t)−

(σ + d2A + γr

)e2A(t)

∂tS(t) = α(%(t)− S(t)− e2A(t)

)e1(t)− β S(t) e2D(t)

+(

(1− θr) (σ + d2A)− γ%)e2A(t)− (γr + γ%)S(t)

∂t%(t) = r − q e2A(t)− γ% %(t)∂tp(t) = θp q e2A(t)− γ n(t) p(t)− γp p(t)

∂tn(t) =µn(t)

n̄+ n(t)

(γ p(t) + (1− θp) q e2A(t)

)− γn n(t).

(3.24)

Remark 3.2. Note that system (3.24) is obtained by reduction under the assumption that ratesare independent of the length of the chain and the number of landing sites. It is a closed systemof seven ODE’s which still keeps the important cascading structure of enzymes acting one afteranother on the cellulose. However, this model assumes that the topology of the cellulose chains(their length and the location of landing sites) does not affect the biodegradation process.

14

Remark 3.3. We make the assumption that the coefficients are constant to derive the S-system.Although this assumption is clearly non-physical, it yields a useful model. We justify dropping theextra correction term which is small relative to other terms, in the expectation that the mathemat-ical error in doing so is much smaller than the modeling errors made by taking the mean coefficient;this is justified by the numerical experiments in Section 6.

3.2.1 S-system with fast transitions of e2A and e2D

In this section we will get a modified version of the S-system given by (3.24) assuming that thetransitions e2D → e2A and e2A → e2D are fast.

Recall that βS is the rate of transition of e2D to e2A and σ is the rate of transition to e2D.Thus, if one assumes that β, σ → ∞ so that βσ stays bounded, then the equation (3.24)3 tends tothe equilibrium relation

0 = (β/σ)S(t) e2D(t)− e2A(t) or e2A(t) = ωS(t) e2D(t) with ω :=β

σ. (3.25)

Thus, the total number of enzymes e2 can be written as

e2 = e2D + e2A = e2D + ωSe2D = e2D(1 + ωS),

so we obtain

e2D = e2RD(ωS) and e2A = e2RA(ωS) where RD(x) :=1

1 + x, RA(x) :=

x

1 + x. (3.26)

Assume next that d2A = d2D. Then the total number of enzymes e2 satisfies the equation

∂te2 = b2n− d2e2 − γre2A = b2n− (d2 + γrRA(ωS))e2 (3.27)

where we have set d2 = d2A = d2D.Thus, replacing (3.24)2,3 with (3.25) and (3.27) we obtain the system, called the S2 system,

∂te1(t) = b1 n(t)− d1 e1(t)∂te2(t) = b2 n(t)− (d2 + γrRA(ωS))e2

∂tS(t) = α(%(t)− S(t)− e2RA(ωS)

)e1(t)− β S(t) e2RD(ωS)

+(

(1− θr) (σ + d2A)− γ%)e2RA(ωS)− (γr + γ%)S(t)

∂t%(t) = r − q e2RA(ωS)− γ% %(t)∂tp(t) = θp q e2RA(ωS)− γ n(t) p(t)− γp p(t)

∂tn(t) =µn(t)

n̄+ n(t)

(γ p(t) + (1− θp) q e2RA(ωS)

)− γn n(t).

(3.28)

The system (3.28) is a version of the S-system with fast transitions of e2A and e2D.

15

3.3 Cleaving Mechanism 2: T -model

We now modify the S2-system derived for the cleaving Mechanism 1 to a model derived for thecleaving Mechanism 2. We directly work from the already reduced S2-system (3.28). Note thatit is of course possible to derive this model from a more fundamental model in which we directlyimplement a corresponding cleaving mechanism on chains of length l, similar to our derivation ofthe N -S model above.

In Mechanism 2, we do not distinguish between attached and detached enzymes e2. In addition,we do not distinguish between occupied and unoccupied sites, preferring to count the total numberof available landing sites T (t), which satisfies (2.1). Our goal is to rewrite the S2-system (3.24) interms of T (t) rather than S(t).

Recall that the S2-system (3.28) is a version of the S-system obtained under the assumptionthat transitions of the enzyme e2 happen fast, which is equivalent to the requirement (3.25). Thus,adding (3.24)3 and (3.24)4 and using the equilibrium relation (3.25), we conclude that the totalnumber of landing sites T = S + e2A (with fast e2 transitions) satisfies the equation

∂tT (t) = α(%(t)− T (t)

)e1(t)− θr (σ + d2) e2(t)RA(ωS)− (γr + γ%)T (t) , (3.29)

where we employed the relationship e2A = e2RA(ωS) and e2D = e2RD(ωS), and the assumptiond2 = d2D = d2A.

We next consider the action of the enzyme e2 in the Mechanism 2. In this model, the concen-tration of e2A enzymes is always low as those enzymes are used and degraded immediately afterthey are produced. The landing, cleaving off of a cellobiose unit, and detaching all occur at approx-imately the same instant. Thus all enzymes e2 should be considered unattached, so should mostclosely resemble e2D. To model this scenario we consider the S2-system in the asymptotic regimein which

ω =β

σ→ 0 and qω = qβ

σ→ q̂ . (3.30)

The first assumption makes sure that the enzyme e2 detaches immediately after the attack and asa consequence the proportion between e2A and e2D tends to zero, while the second one is necessaryfor the population n(t) to survive.

Observe that under assumptions (3.30)

R2A(ωS) ∼ ωS , T = S + e2RA(ωS) ∼ S , (σ + d2)e2RA(ωS) ∼ βe2T , qe2RA(ωS) ∼ q(β/σ)e2T

and so the S2-system transforms into the T -system for Mechanism 2:

∂te1(t) = b1 n(t)− d1 e1(t)∂te2(t) = b2 n(t)− d2e2(t)∂tT (t) = α

(%(t)− T (t)

)e1(t)− θr βe2(t)T (t)− γ̂T T (t)

∂t%(t) = r − q̂ T (t)e2(t)− γ% %(t)∂tp(t) = θp q̂ T (t)e2(t)− γ n(t) p(t)− γp p(t)

∂tn(t) =µn(t)

n̄+ n(t)

(γ p(t) + (1− θp) q̂e2(t)T (t))

)− γn n(t).

(3.31)

where we have set q̂ = q (β/σ) and γ̂T = γr + γ%.

16

Comparing this system to (3.24) modeling Mechanism 1, we make the following observations:first, the dynamics of e2 and T are simpler than those of e2A, e2D and S, because we do nothave to account for the different processes for e2D and e2A. On the other hand, the cleaving ratechanges from the linear term q e2A in the S-system, to the nonlinear term q̂ T e2 which modelssimultaneously finding and attacking a landing site. Note that the coefficient q̂ = q (β/σ) combinesthose coefficients corresponding to finding sites and cleaving off in Mechanism 1, while γ̂T = γr+γ%is the combined rate of degradation of landing sites.

3.4 Multiple trait T -model

We now extend the T -model to a model that allows for several species of microorganisms feedingon the same cellulose. Specifically, we introduce populations ni, with i ∈ {1, ...M}, equipped withdifferent traits xi; throughout this section, we use superscripts to distinguish traits. We assumethat Mechanism 2 is used by each population to cleave off cellobiose, while the different populationshave different rates for the various actions.

As in the T -model, we assume that microorganism ni produces endoglucanases enzymes ei1 that

make landing sites, and exoglucanases enzymes ei2 that cleave off cellobiose from the cellulose chains.In analogy with (3.31)1,2, we suppose that the enzymes e

i1, e

i2 are produced by the microorganism

ni and degrade with fixed rates. This gives the equations

∂tei1(t) = b

i1 n

i(t)− di1 ei1(t)∂te

i2(t) = b

i1 n

i(t)− di2 ei2(t)(3.32)

where bi1, bi2 and d

i1, d

i2 are, respectively, the enzyme generation and death rates, i ∈ {1, ...M}.

Next, for simplicity we will not differentiate between landing sites created by the enzymes ofdifferent species. In other words, the landing sites made by ei1 are allowed to be used by anyenzyme ej2 for all j. Then, following Mechanism 2, and neglecting saturation effects, we assumethat enzymes ei1 make landing sites on the cellulose % with the rate

αi(%(t)− T (t)

)ei1(t),

where αi is the probability of an individual enzyme ei1 finding a spot among the %(t)− T availablecellobiose units (reflecting the requirement that only one landing site per cellobiose unit is allowed),and making a landing site. Next, we suppose that the landing sites T are attacked by enzymes ei2with the rate

βi T (t) ei2(t)

where βi is the probability of an individual ei2 finding and attaching to a landing spot. Finally, asabove, we let γr > 0 be the decay rate of an individual landing site and suppose that the portionθir ∈ [0, 1] of those ends is not usable after an attack by ei2. Then, in analogy with (3.31)3, weobtain the equation for T for multiple trait populations:

∂tT (t) =∑j

αj(%(t)− T (t)

)ej1(t)−

∑j

θjr βj T (t) ej2(t)− γ̂T T (t). (3.33)

Next, we let qi be the number of cellobiose units cleaved off by ei2 during an attack. In analogywith (3.31)4, the dynamics of cellulose % is then given by

∂t%(t) = r −∑j

q̂j T (t) ej2(t)− γ% %(t) (3.34)

17

where we set q̂i = qi βi, the combined rate of attack of ei2.We next let θip ∈ [0, 1] denote the proportion of produced cellobiose produced by ei2 that is

made available to everyone. Then, as in (3.31)5, the equation for the total amount p(t) of cellobioseavailable to everyone is

∂tp(t) =∑j

θjp q̂j ej2(t)T (t)−

∑j

γj nj(t) p(t)− γp p(t) . (3.35)

where γj is the predation rate of p by nj , and γp is the decay rate of p.Finally, we consider the dynamics of the population ni. First, recall that cellobiose p(t) is

available to all species nj , j = 1, . . . ,M . Since every species nj hunts with the predation rate γj

on the cellobiose p, the growth rate of ni may be expressed via the logistic term

µi ni(t)γi p(t)

n̄i + 1γi

∑j γ

j nj(3.36)

where µi is the conversion efficiency.Next, comparing to (3.35), the production rate of cellobiose which is produced by ej2 and con-

sumed directly on the spot is given by

(1− θjp) q̂j ej2(t)T (t).

We assume that in view of the homogeneity and close proximity of species, the cellobiose producedby ej2, j = 1, . . . ,M , can be consumed by the n

i; this manifests the cross species interaction. Weexpress this as

µi (1− θjp)νij ni

νij n̄i +∑

s νsjns

q̂j ej2(t)T (t), with∑s

νsj = 1,

representing the contribution of the energy obtained from the direct consumption of cellobiosecleaved off by ej2 to the growth rate of n

i. Combining these leads to the equation for the dynamicsof the population ni,

∂tni(t) = µini(t)

γip(t)n̄i + 1

γi

∑j γ

j nj+∑j

(1− θjp) νij

νij n̄i +∑

s νsj ns

q̂j ej2(t)T (t)

− γin ni(t) (3.37)where γin is the death rate of the population n

i.

18

We collect the above equations to obtain the multiple trait T -system,

∂tei1(t) = b

i1 n

i(t)− di1 ei1(t)∂te

i2(t) = b

i2 n

i(t)− di2 ei2(t)

∂tT (t) =(%(t)− T (t)

) ∑j

αj ej1(t)−∑j

θjrβj ej2(t)T (t)− γ̂TT (t)

∂t%(t) = r −∑j

q̂j ej2(t)T (t)− γ% %(t)

∂tp(t) =∑j

θjp q̂j ej2(t)T (t)−

∑j

γj nj(t) p(t)− γp p(t)

∂tni(t) = µi ni(t)

γi p(t)

n̄i + 1γi

∑j γ

j nj

+ µini(t)∑j

(1− θjp)νij

νijn̄i +∑

s νsj ns

q̂j ej2(t)T (t)− γin n

i(t)

(3.38)

where i = 1, . . . ,M .

3.4.1 Compatibility with the single trait model

We now show that the multiple trait T -model directly generalizes the single trait T -model byconsidering two special cases of the multiple trait model, and confirming that these reduce to thesingle trait model.

Our first test is to assume that all but one species (say the i-th) are absent. That is, we beginwith data nj(0) = 0, and similarly ej1(0) = e

j2(0) = 0, for j 6= i. Then (3.38) implies that for

all t > 0, j 6= i, we have nj(t) = 0. It is then evident that equations (3.38)1,2,3,4,5 reduce to(3.31)1,2,3,4,5 (for n = n

i, etc), and (3.38)6 becomes

∂tni(t) = µi ni

γi p

n̄i + ni+ µini (1− θip)

q̂i ei2 T

n̄i + ni− γin ni,

which is exactly (3.31)6.Our second test is to assume that even though there are M different traits, the coefficients are

independent of i, j, so there is no way to distinguish different populations in the model. In thiscase, we check the dynamics for the total population n(t) =

∑i n

i(t), and similarly e1 =∑

i ei1 and

e2 =∑

i ei2. It is then clear that (3.38)3,4,5 become (3.31)3,4,5, and adding (3.38)1,2 over i gives

(3.31)1,2. Finally, adding (3.38)6 over i yields

∂t(∑

ini) = µ

∑ini γ p

n̄+∑

jnj

+ µ∑

ini 1− θpn̄+

∑snsq̂∑

jej2 T − γn

∑ini,

which is again exactly (3.31)6.

19

4 Cooperation

4.1 Cooperation in the T -model

In this section we consider the system (3.31) on different time scales. We assume that productionof enzymes, consumption and creation of landing sites occurs at a much faster rate than changesin the population of the microorganism. In this case, over time scales on which the populationchanges, we can assume that equations (3.31)1,2,3,4,5 are at equilibrium, and the dynamics is drivenby the population change (3.31)6. This gives the system

0 = b1 n(t)− d1 e1(t)0 = b2 n(t)− d2 e2(t)0 = α

(%(t)− T (t)

)e1(t)− θr β T (t) e2(t)− γ̂T T (t)

0 = r − q̂ T (t) e2(t)− γ% %(t)0 = θp q̂ T (t) e2(t)− γ n(t) p(t)− γp p(t)

∂tn(t) =µn(t)

n̄+ n(t)

(γ p(t) + (1− θp) q̂ T (t) e2(t)

)− γn n(t).

(4.1)

We use (4.1) to eliminate all the fast variables, to obtain a single equation for the populationn, so that the population growth rate can be understood. The first two equations give

ei = ki n with ki :=bidi, i = 1, 2 , (4.2)

and from (4.1)4, we have

% =r

γ%− q̂γ%T e2 =

r

γ%− q̂ k2

γ%T n.

Plugging these into (4.1)3, we get

0 =α r k1γ%

n− T(α q̂ k2 k1

γ%n2 + (αk1 + θrβ k2)n+ γ̂r

),

so that

T =α r k1γ%

n

P2(n), (4.3)

where P2(n) is the quadratic polynomial

P2(n) = c2 n2 + c1 n+ c0,

c2 =α q̂ k2 k1γ%

, c1 = αk1 + θr β k2, c0 = γ̂r.(4.4)

Next, using these in (3.31)5, it follows that

p = θp q̂ k2T n

γ n+ γp= θp q̂ k2

α r k1γ%

n2

(γ n+ γp)P2(n). (4.5)

Finally, we use (4.3), (4.5) in (4.1)6 to get the scalar population equation

∂tn(t) = n(t)[B(n)− γn

], (4.6)

20

where

B(n) =µ

n̄+ n

(γ p+ (1− θp) q̂ k2 T n

)= µ q̂ k2

T n

n̄+ n

(γ θp

γ n+ γp+ (1− θp)

)= K n2 Φ(n),

(4.7)

where the function Φ(n) and constant K are given by

Φ(n) =

(θp

n+ γp/γ+ 1− θp

)1

(n+ n̄)P2(n),

K = µ q̂ k2α r k1γ%

=µ q β α r b1 b2γ% d1 d2

.

(4.8)

4.2 Asymptotics of B(n)

We are interested in the structure of B(n) for small populations, n � n̄. First, we note that thebirth rate B(n) is positive and satisfies

B(0) = 0 and limn→∞

B(n) = 0,

so that B is globally bounded.For n small, using (4.7), (4.8) and (4.4), we have

B(n)

n2= K Φ(n) ≈ K Φ(0) = K

n̄ γ̂r

(θp

γ

γp+ 1− θp

), (4.9)

so that B(n) ∼ n2 for n� n̄. Thus for small populations, B(n) is convex, and so superlinear. Thissuperlinear birth rate is indicative of cooperative behavior.

More generally, the growth rate B(n)/n increases as long as log(nΦ(n)

)does, and

∂

∂n

(B(n)n

)=

∂

∂n

(K nΦ(n)

)= K nΦ(n)

( 1n

+∂nΦ

Φ

)= K nΦ(n)

( ∂∂n

log(nΦ)),

so the system exhibits cooperative behavior as long as

∂

∂nlog(nΦ) =

1

n− θp

(n+ γp/γ)2

(θp

1

n+ γp/γ+ 1− θp

)−1− 1n+ n̄

− 2c2n+ c1P2(n)

> 0 .

This condition is at least true for n ∈ (0, n∗), where n∗ is the smallest positive root of this expression;combining the fractions, it is evident that n∗ is the smallest positive root of a fifth-order polynomial.

Moreover, referring to (4.9), we see that

∂

∂θpK Φ(n)

∣∣∣n=0

=K

n̄ γ̂r

( γγp− 1),

which is positive if and only if γ > γp. For small population n, we expect this to persist: that is,

∂

∂θp

(B(n)n2

)=

∂

∂θpK Φ(n) > 0 if and only if γ > γp.

Since θp ∈ [0, 1] is the proportion of produced cellulose which is shared, this last inequality suggeststhat for small populations sharing food is beneficial in terms of growth as long as the consumptionrate γ is greater than the decay rate γp of the cleaved off cellobiose.

21

4.3 Cooperation in multiple-trait T -model

As in Section 4.1, we consider the system (3.38) on a generational time scale. We again assumethat production of enzymes, consumption and creation of landing sites happens at a much fasterrate then change in the populations ni. In other words, we assume that the equations (3.38)1,2,3,4,5are at equilibrium, and the dynamics of the system is driven by the population equations (3.38)6.This results in the system

0 = bi1 ni(t)− di1 ei1(t),

0 = bi2 ni(t)− di2 ei2(t),

0 =(%(t)− T (t)

) ∑j

αj ej1(t)−∑j

θjr βj ej2(t)T (t)− γ̂T T (t),

0 = r −∑j

q̂j ej2(t)T (t)− γ% %(t),

0 =∑j

θjp q̂j ej2(t)T (t)−

∑j

γj nj(t) p(t)− γp p(t),

∂tni(t) = µi ni(t)

γip(t)

n̄i + 1γi

∑j γ

j nj,

+ µi ni(t)∑j

(1− θjp)νij

νij n̄i +∑

s νsj ns

q̂j ej2(t)T (t)− γin n

i(t).

(4.10)

We wish to understand the growth rate of ni as a function of n = (n1, . . . , nM ) ∈ RM . The firsttwo equations of (4.10) give

ei1(t) = ki1n

i(t), ei2 = ki2n

i, with ki :=bi

di, i = 1, . . . ,M . (4.11)

We next write n as a vector and introduce the coefficient vectors

A =(αj kj1

), B =

(θjr β

j kj2

), Q =

(q̂j kj2

),

Θ =(θjp q̂

j kj2

), Γ =

(γj), Nk =

(νjk),

(4.12)

and denote the scalar product by 〈·, ·〉. We can then rewrite (4.10)3,4,5 as

0 =(%(t)− T (t)

)〈A,n〉 − 〈B,n〉T (t)− γ̂T T (t),

0 = r − 〈Q,n〉T (t)− γ% %(t),0 = 〈Θ, n〉T (t)− 〈Γ, n〉 p(t)− γp p(t).

These immediately yield

%(t) =1

γ%

(r − T (t) 〈Q,n〉

)and p(t) =

〈Θ, n〉〈Γ, n〉+ γp

T (t), (4.13)

and, plugging in, we get

0 =r

γ%〈A,n〉 − τ(n)T (t), so that T (t) = r

γ%

〈A,n〉τ(n)

, (4.14)

22

where we have set

τ(n) =1

γ%〈A,n〉 〈Q,n〉+ 〈A,n〉+ 〈B,n〉+ γ̂T . (4.15)

Finally, using (4.13) and (4.14) in (4.10)6, we can write our population system as

∂tni = ni

(Bi(n)− γin

), (4.16)

where the i-th population’s birth rate is

Bi(n) =µi γi p(t)

n̄i + 1γi〈Γ, n〉

+ µi∑j

(1− θjp) νij q̂j kj2 nj

νij n̄i + 〈N j , n〉T (t)

= µi T (t)

(γi 〈Θ, n〉

(n̄i + 1γi〈Γ, n〉) (〈Γ, n〉+ γp)

+∑j

(1− θjp) νij q̂j kj2 nj

νij n̄i + 〈N j , n〉

)

=µi r

γ%

〈A,n〉τ(n)

(γi 〈Θ, n〉

(n̄i + 1γi〈Γ, n〉) (〈Γ, n〉+ γp)

+∑j

(1− θjp) νij q̂j kj2 nj

νij n̄i + 〈N j , n〉

).

(4.17)

Here the two terms in the growth rate correspond to intentionally shared food and food consumedas it’s produced, respectively.

Asymptotic behavior of Bi(n). Assuming the coefficients are positive, we make the followingobservations about the birth rate Bi(n). According to (4.15) τ(n) is quadratic in n, while all innerproducts in (4.16) are linear. It follows immediately that Bi(n)→ 0, and in fact Bi(n) = O( 1n) asn→∞.

We are more interested in the behavior for small populations, n ∼ 0. Since τ(0) = γ̂T , nodenominators vanish, and (4.17) yields

Bi(n) = O((∑n)2)

for n ∼ 0.

More precisely, recalling that

∇n〈V, n〉 = V and D2n(〈V, n〉 〈W,n〉

)= V W T +W V T ,

we see that at n = 0, the gradient of Bi vanishes, ∇nBi(0) = 0, and the Hessian of Bi is thesymmetric matrix

D2nBi(0) =

µi r

γ% γ̂T n̄i

(γi

γp(AΘT + ΘAT ) +A(Q−Θ)T + (Q−Θ)AT

).

We cannot conclude that Bi is convex as the matrix D2nBi(0) is not positive definite, but because

all the entries are positive, we can conclude that the directional derivative is increasing in anydirection in the positive orthant {nk ≥ 0}, which indicates cooperative behavior.

Special Case. Now, we consider the special case when νij = 0 for i 6= j; in this case, there is nocompetition for cellobiose that is not intentionally shared. In this special case, (4.17) becomes

Bi(n) =µi r

γ%

〈A,n〉τ(n)

(γi 〈Θ, n〉

(n̄i + 1γi〈Γ, n〉) (〈Γ, n〉+ γp)

+(1− θip) q̂i ki2 ni

n̄i + ni

)=: Bi1(n) +B

i2(n).

23

As in the single-trait case, we again see an indication that for small populations, more sharing(represented by the coefficient vector Θ) would be beneficial for the i-th population provided γi >γp, because it increases the derivative ∇nBi(n): this can be seen by differentiating with respect tothe vector parameter Θ. Recall that γi > γp means that cellobiose is consumed (by n

i) faster thanit decays.

Lemma 4.1. Suppose that νij = 0 for i 6= j. Let i ∈ {1, . . . ,M} be fixed and let

n0 = (n10, n

20, . . . , n

i−10 , 0, n

i+10 , . . . , n

M0 ) ∈ RM with n

j0 ≥ 0.

Then∂Bi2∂ni

(n0) =µi r

γ%

〈A,n0〉τ(n0)

(1− θip) q̂i ki2n̄i

> 0. (4.18)

Furthermore, suppose minj ᾱj > 0 and minj γ

j > 0. Then there exists ε > 0 such that for all

maxj θjp < ε, we have

∂Bi

∂ni(n0) > 0 for all n0 6= 0 .

Idea of proof. Equation (4.18) follows immediately by differentiation. When we differentiate Bi1,we introduce negative terms each time the derivative falls on a denominator. However, each suchterm introduces a higher power in the denominator, so each of those terms can be represented asa product of 〈A,n0〉/τ(n0) with terms uniformly bounded in n0. Comparing these to (4.18), itfollows that by choosing maxj θ

jp < ε with ε small enough, the sum will be positive.

Remark 4.1. The key conclusion in Lemma 4.1 is that the birth rate Bi(n) includes some formof cooperation. Namely, the cooperative effect is not eliminated in the multiple trait populationT -model provided that νij = 0 for i 6= j. In other words, when interspecies competition for ‘readyto be digested’ resource p is not involved then there is an Allee effect for each species. Compareit for instance with simple logistic terms like r − ni0 which decreases with ni0. In contrast Bi(n)actually penalizes populations which are too small (and populations which are too large of coursejust like a logistic term). Finally, even if the conditions of Lemma 4.1 do not hold a cooperativeeffect still maybe present depending on the differential DB(0).

4.4 Cooperative interactions in N-S-model

We now consider cooperation in the N -S-model as we did for the simpler models. We are interestedin the situation that the production of enzymes, consumption and creation of landing sites occursmuch faster than changes in the population of the microorganism. Thus we again assume thatequations (3.3), (3.5), (3.11) and (3.12)1 are at equilibrium, and the dynamics is driven by thepopulation change (3.12)2. We also assume that the length of the cellulose chains does not exceed

24

a given number L > 0. Then we obtain the system

0 = b1n(t)− d1e1(t)

0 = b2n(t)−∑l,i

[βl,i(iN l,i(t)− el,i2A(t)

)e2D(t)− σl,iel,i2A(t)

]− d2D e2D(t)

0 = βl,i(iN l,i(t)− el,i2A(t)

)e2D(t)−

(σl,i + d l,i2A + γ

l,ir

)el,i2A(t)

0 = rl,i +(α̂l,i−1N l,i−1(t)− α̂l,iN l,i(t)

)e1(t) +

(γ̂l,i+1N l,i+1(t)− γ̂l,iN l,i(t)

)+(ql+1,iel+1,i2A (t)− q

l,iel,i2A(t))

+ δliql+1,l+1el+1,l+12A

+(θ̂l,i+1el,i+12A (t)− θ̂

l,iel,i2A(t))− γl,i% N l,i(t)

0 = θp∑l,i

ql,iel,i2A(t)− γn(t)p(t)− γpp(t)

∂tn(t) =µn(t)

n̄+ n(t)

(γ +

1− θpθp

(γn(t) + γp))p(t)− γnn(t) ,

(4.19)

for (l, i) ∈ IL, where δli is the Kronecker delta, the rates rl,i = 0 when i 6= 0 and we use theconvention (3.2); here we have used (4.19)5 to simplify (4.19)6.

To describe the dynamics, it is sufficient to express p in terms of n, which will in turn providea scalar autonomous differential equation for n(t). For small populations the equations (4.19)1−5can be solved uniquely in terms of n, yielding the following theorem.

Theorem 4.1. There are m, m̄ > 0 and C∞ functions

ê1(n), ê2D(n), êl,i2A(n), N̂

l,i(n), p̂(n) : (−m, m̄)→ R (4.20)

such that:

(i) For each n ∈ (−m, m̄) the equations (4.19)1−5 can be solved uniquely for e1, e2D, el,i2A, N , pin terms of n,

e1 = ê1(n), e2D = ê2D(n), el,i2A = ê

l,i2A(n), N

l,i = N̂ l,i(n), p = p̂(n).

(ii) The functions from (4.20) are given to leading order as

N̂ l,i(n) = νl,ini +O(ni+1) , (4.21)

with

νl,0 =rl,0

γl,0%and νl,i = νl,i−1

α̂l,i−1

γ̂l,i + γl,i%

b1d1, i ≥ 1 ,

and with

25

ê1(n) =b1d1n, ê2D(n) =

b2d2D

n+O(n2)

êl,i2A(n) = i

(b2β

l,iνl,i

d2D(σl,i + dl,i2A + γ

l,ir )

+O(n)

)ni+1

γpθpp̂(n) = p̄(n)n2, where p̄(n) =

b2d2D

∑l

ql,1βl,1νl,1

(σl,1 + d l,12A + γl,1r )

+O(n) . (4.22)

Proof. For fixed t, we regard (4.19)1-(4.19)5 as an algebraic system, which can be solved uniformlyin t, for small n. First, (4.19)1 yields

e1 = k1 n, with k1 =d1b1,

and subtracting all of (4.19)3 from (4.19)2 and solving gives

e2D =b2d2D

n−∑l,i

d l,i2A + γl,ir

d2Del,i2A. (4.23)

Next, (4.19)5 yields

p =θp

γn+ γp

∑l,i

ql,i el,i2A,

while (4.19)3 gives

el,i2A = iNl,i

(1 +

σl,i + d l,i2A + γl,ir

βl,i e2D

)−1= ηl,iN l,i, (4.24)

where we have set

ηl,i = i(

1− 11 + ζ l,i e2D

)= i ζ l,i e2D

(1 +O(e2D)

), with

ζ l,i =βl,i

σl,i + d l,i2A + γl,ir

.

Now, regarding n and e2D as fixed, we use (4.24) in (4.19)4 to get

0 = rl,i +(α̂l,i−1N l,i−1 − α̂l,iN l,i

)k1 n+

(γ̂l,i+1N l,i+1 − γ̂l,iN l,i

)+(ql+1,i ηl+1,iN l+1,i − ql,i ηl,iN l,i

)+ δl,i q

l+1,l+1 ηl+1,l+1N l+1,l+1

+(θ̂l,i+1 ηl,i+1N l,i+1 − θ̂l,i ηl+1,iN l,i

)− γl,i% N l,i,

(4.25)

which we regard as a linear system,

AN = r, for NT =(N1,0, N1,1, N2,0, N2,1, N2,2, N3,0, . . . NL,L

).

26

When expressed in matrix form, the matrix A is sparse and upper Hessenberg, with subdiagonalentries −α̂l,i−1k1n. It follows that the matrix is upper triangular for n = 0, so invertible for smalln, and we get a unique solution N l,i = N l,i(n, e2D).

Setting n = e2D = 0, and recalling that rl,i = 0 for i > 0 and γ̂l,0 = 0, (4.25) gives the initial

solution

N l,0(0, 0) =rl,0

γl,0%, N l,i(0, 0) = 0, i > 0,

and so using (4.24) and (4.23), we get in particular el,i2A = ηl,i = 0 whenever n = e2D = 0.

We now plug the solution N l,i = N l,i(n, e2D) into (4.24), (4.23) to get

G(n, e2D) := e2D −b2d2D

n+∑l,i

d l,i2A + γl,ir

d2Dηl,iN l,i(n, e2D) = 0,

relating e2D to n. We calculate∂G

∂e2D

∣∣∣(0,0)

= 1,

so the implicit function theorem implies that, for n small enough, there is a unique function e2D(n)such that G(n, e2D(n)) = 0, and moreover,

e2D =b2d2D

n+O(n2).

Finally, we have

N l,i(n) = N l,i(0, 0) +O(n) and ηl,i = ib2 ζ

l,i

d2Dn+O(n2),

so according to (4.24), we have

el,i2A = O(n2), so also p = O(n2).

Moreover, for small n, we can write (4.25) for i ≥ 1 as

0 =α̂l,i−1 k1 nNl,i−1 −

(γ̂l,i + γl,i% +O(n)

)N l,i

+(γ̂l,i+1 +O(n)

)N l,i+1 + δl,i q

l+1,l+1 ηl+1,l+1N l+1,l+1,

and we can solve this inductively in i, to get

N l,i =α̂l,i−1 k1

γ̂l,i + γl,i%N l,i−1 n

(1 +O(n)

),

from which the result follows.

Birth rate B[n]. By Theorem 4.1, and using (4.19)6 and (4.22), we conclude that for smallpopulations n ∈ [0, m̄), the dynamics is again driven by the equation

∂tn(t) = n(t)(B(n)− γn

),

27

where now the birth rate B(n) is given by

B(n) =µ

n̄+ n(t)

(γ +

θp − 1θp

(γn+ γp))p̂(n)

=µn2

n̄+ n(t)

( γγpθp +

(1− θp

)(γ

γpn+ 1)

)p̄(n) .

(4.26)

We now divide B(n) by n2 and differentiate with respect to θp. Since p̄ is independent of θp,we obtain

∂

∂θp

(B(n)

n2

)∣∣∣∣n=0

=µ

n̄

( γγp− 1) b2d2D

mc∑l

ql,1βl,1νl,1

(σl,1 + d l,12A + γl,1r )

,

so that∂

∂θp

(B(n)

n2

)∣∣∣∣n=0

> 0 if and only if γ > γp .

Thus we arrive at a similar conclusion to that of the T -model: that is, for small populations,sharing food (within the species) is beneficial in terms of growth as long as the consumption rateγ is greater than the decay rate γp of the cleaved off cellobiose.

5 A Model in the Continuous Setting

In this section, by analogy with our multiple trait T -model (4.16), we develop a model for apopulation with any number of traits, which we write for convenience in the continuous setting.We should say again that in our context, we do not expect the number of possible enzymes forcellulose degradation to be that large but note that this model includes finite trait models byappropriate use of δ-functions,

n(t, x) =∑j

nj(t) δxj (x).

We hence present the continuous model here for its generality and because the resulting equationmay be more amenable to analysis.

We think of the multiple-trait population as having M traits indexed by x1 < · · · < xM , so wewrite

ni(t) = n(xi, t)∆x, ei1(t) = e1(xi, t), ei2(t) = e2(x

i, t),

for some functions n(x, t), e1(x, t), e2(x, t), representing the population and enzyme densities. Wenow simply assume that the variable x takes on a continuous range of values.

We similarly translate the coefficient vectors (4.12), so that these become continuous parameters:that is, we allow the parameters bi, di, α, β, q̂, θp, etc, to depend on x, and in analogy to (4.12)1,2,3we set

A(x) = α(x)b1(x)

d1(x), B(x) = θr(x)β(x)

b2(x)

d2(x), Q(x) = q̂(x)

b2(x)

d2(x), (5.1)

where these are now positive functions. In particular, we interpret α, β and q̂ as the rates of landingsite generation, occupation, and the rate of cellobiose production, per individual, respectively.

28

We now simply follow the development that led to (4.16), but reinterpreting the inner product,so that for each function W (x),

〈W,n〉 =∫W (y)n(y, t) dy.

Then (4.13), (4.14) and (4.15) are unchanged. To express the population equation, we define thefunctional

τ [n] :=1

γ%mc〈A,n〉〈Q,n〉+ 〈A,n〉+ 〈B,n〉+ γp,

and the convolution

N [n](x, t) =

∫ν(s, x)n(s, t) ds,

which is an inner product in the first variable.We must now model the last term in (4.17). In analogy with that term, we define

Ξ(z;x, t) =ν(x, z)Q(z)

ν(x, z) n̄(x, t) +N [n](z, t).

Now, in analogy with (4.16), (4.17), we write the population equation as

∂tn(x, t) = n(x, t)(B[n](x, t)− γn(x)

),

where the birth rate is now the functional

B[n] =µ(x) r

γ%mc

〈A,n〉τ [n]

(〈θpQ,n〉γ(x)(

n̄(x) + 1γ(x) 〈γ, n〉) (〈γ, n〉+ γp

) + 〈(1− θp) Ξ(· ;x, t) , n〉) . (5.2)6 Numerical experiments

In this section we use numerical experiments to test and compare the various models that we havederived.

N -S and S models. First, we compare the N -S-model with the S-model. The results of thenumerical computations are presented in Figure 2. The coefficients of the S-system are chosen as

b1 = 0.5, b2 = 0.5, d1 = 0.5, d2A = 0.5, d2D = 0.5

β = 0.5, σ = 0.1, γr = 0.01, γp = 0.001, γ% = 0.001

γn = 0.1, α = 0.05, r = 1000, θr = 0.05, θp = 0.75

q = 1, γ = 0.005, µ = 0.5, n̄ = 100.

(6.1)

The coefficients of the N -S-system are chosen randomly as follows. For each coefficient of theS-system, let us call it ‘c’, the corresponding coefficient cl,i of the N -S-system (which explicitlydepends on the state of the chain (l, i)) is chosen as

cl,i = cX where X ∼ gamma(k, θ), k = p−2, θ = p2 . (6.2)

Here the value p is the standard deviation of X. Thus all samples approximately lie in (c−3p, c+3p).

29

Figure 2: Solutions of S and NS systems with 3p = 0.05

S and T models. We next compare the T and S systems numerically. The coefficients of theS-system are chosen as

b1 = 0.1, b2 = 0.1, d1 = 0.1, d2A = 0.1, d2D = 0.1

γr = 0.01, γp = 0.005, γ% = 0.005, γn = 0.01, α = 0.05

r = 1000, θr = 0.05, θp = 0.75, γ = 0.01, µ = 0.5,

n̄ = 100

(6.3)

and{β}6i=1 = {0.6668, 0.8394, 1.0567, 1.3304, 1.6748, 2.1085}{σ}6i=1 = {17.7828, 177.828, 1778.28, 17782.8, 177828, 1778280}{q}6i=1 = {0.0007, 0.0053, 0.0421, 0.3342, 2.6544, 21.0848}.

(6.4)

The pictures on Figure 3 correspond to the limiting procedure where

qiβiσi

= q0 = 0.000025 andβiσi→ 0.

30

(a) i = 1, βiσi = 0.11857 (b) i = 2,βiσi

= 0.03749 (c) i = 3, βiσi = 0.01185

(d) i = 4, βiσi = 0.00374 (e) i = 5,βiσi

= 0.00118 (f) i = 6, βiσi = 0.00037

Figure 3: T (t) of T -model and S-model for qiβiσi

= q0 = 0.000025 andβiσi→ 0.

7 Appendix

7.1 Tail issue in deterministic selection dynamics

Models in population dynamics focus on selection because it is rightfully viewed as the main mecha-nism to explain the survival of populations and the evolution of traits. The selection mechanism inthese models is often driven by competition between individuals, possibly combined with mutationsto create new traits. In addition competition is well understood from the modeling point of view.

On the other hand cooperative effects are harder to model, especially at the level of micro-organisms. Several well-known cooperative effects (such as sexual reproduction for large animals)do not take place for all micro-organisms. Nevertheless, the importance of such effects has longbeen recognized: see for instance the works [14, 23, 24, 28] on mutualism that discuss interspeciesinteractions yielding reciprocal benefits.

In this paper we introduce biological mechanisms, by the example of cellulose bio-degradation,

31

that lead to reproduction rates encoding both (intra-species) cooperative effects and competitionbetween individuals; see Section 4. This suggests that reproduction rates that only incorporatecompetition may fail to describe many biochemical processes, especially at the level when B[n]significantly deviates from traditional logistic terms, that is for small populations.

There are several approaches to study the phenotypical evolution driven by small mutations inreplication, the main objective being to describe the dynamics of the fittest (or dominant) traitin the population. The main mechanisms affecting dynamics are usually a) the selection principle(due to competition, birth and death), and b) small mutations. These two mechanisms influencethe trait dynamics on two different scales. The selection effect becomes evident on the reproductiontimescale tR, while the effect of small mutations is evident on a generation timescale tM � tR. Thedrastic difference between the two scales introduces both small and large parameters into models(mutations can be small or rare for instance, population is usually large and death rates could vary)and this causes various difficulties.

One of the best known approaches is the so-called adaptive dynamics theory, see for instance[10, 12, 3, 16, 36]. Adaptive dynamics considers evolution as a series of invasions by a small mutantpopulation of the dominant trait population, a process which is classically modeled by a system ofODE’s. Depending on the relative fitness of the mutant, this can lead to the replacement of thedominant trait or the extinction of the invading population (the cases of co-existence are usuallyharder to handle at this level).

Other very popular models are stochastic, or individual-centered models, see for instance [14, 5,5] among many. Probabilistic models are natural because they take natural fluctuations of births,deaths, and mutations into account at the individual level, and are therefore considered to be themost realistic. They consist in life and death processes for each individual Xi. A typical exampleconsists in taking Poisson processes with birth rates b(Xi) and death rates increasing with thecompetition between individuals, for example di = d(Xi) +

∑j 6=i I(Xi −Xj).

When a birth occurs, it simply adds another individual with the same trait, unless a mutationtakes place, generally with small probability. In that case, the new individual has a different randomtrait, obtained through some distribution. In general of course competition could influence boththe birth rate and the mortality rate. Under the right scalings, stochastic models can lead to theclassical adaptive dynamics [34, 26].

When the total number of individuals is too large (it can easily reach 1010−1012 for some micro-organisms), stochastic models can become cumbersome and prohibitive to compute numerically forinstance. In that case, one expects to be able to derive a deterministic model as a limit of largepopulations which would be simpler to use. Such a derivation was provided in [7] for example,leading to integro-differential equations such as

∂tn(x, t) =(r(x)−

∫I(x− y)n(t, y) dy

)n(x, t) +M [n](t, x), (7.1)

where r(x) = b(x)− d(x) and M [·] is the mutation kernel, a diffusion or integral operator. This isthe level of modeling that we are interested in this article.

Even though deterministic models of type (7.1) are obtained from stochastic ones, simulationsfor these two types of models typically produce different behaviors in terms of evolutionary speedsand branching patterns. In stochastic simulations, in which a single individual represents a minimalunit necessary for survival, demographic stochasticity (the variability in population growth ratesamong individuals) acts drastically on small populations, leading to complete extinction of smallpopulations with negative reproduction rates. In deterministic models however, sub-populations

32

can never go completely extinct and can “rebound” later on if their reproduction rate becomespositive.

It is an open and difficult question of how to keep the stochastic effects for the small populationsin the deterministic models. Perthame and Gauduchon [26] made an attempt in truncating thepopulations with less than one individual by introducing an analog of stochastic mortality formodels of type (7.1), a survival threshold, which allows phenotypical traits in the small populationto vanish in finite time. In [26] this is achieved by modifying (7.1) as follows

∂tn(x, t) =(r(x)− I ? n

)n−

√n

n̄+M [n](t, x) . (7.2)

The new term enables the population to vanish for some traits when the population density is toolow in comparison with n̄, which disallows densities corresponding to fewer than one individual.

As one wishes to see the evolution of traits generated by mutations, one needs to rescale theabove equation in time. This leads to large deviation type phenomena which can be observed bydefining nε(t, x) = exp(φε(t, x)/ε), with ε the ratio of the reproduction and mutation time scales(see [26, 13]). One now has two scales for the populations, the small population threshold n̄ andthe exponential scale expφ/ε.

Often, the aim is to analyze the population behavior in the limit as ε→ 0 and therefore n̄ shouldbe chosen in terms of ε. Numerical simulations for the corresponding equation with initial data ofmonomorphic type, see [26], indicate that the evolution speeds and time of branching depend onthis choice of n̄ in terms of ε. When ε is fixed, too large a value of n̄ leads to extinction, while toosmall a value of n̄ leads to spontaneous jumps in branching, see [26, 13].

A complete mathematical analysis of the general equations is currently intractable. One ofthe few situations that is currently understood [26] is when the mortality threshold is chosen asn̄ε = exp(− ϕ̄ε ). However, the scaling n̄ε = exp(−

ϕ̄ε ) for a fixed ϕ̄ is often much too small. Recall

that the threshold n̄ε should correspond to a single individual in stochastic modeling. Thus, if wecome back to the starting point, which means a total population

∫n(x, t) dx of 1010−1012, then for

ε = 10−4 (a typical value for many applications) and threshold n̄ε of order exp(−1ε ), an aggregatepopulation over any fixed interval of traits would still represent much less than one individual.

Another type of correction has been proposed by Jabin [22]. The author allows the thresholdn̄ε to be polynomial in ε and introduces special cooperative term Dε in the (rescaled) model

∂tnε(x, t) =(r(x)−

∫I(x− y)n(t, y)−Dε[n]

)n(x, t) +M [n](t, x). (7.3)

This term does not handle the small populations as precisely, but the new model still corrects allthe abnormal behaviors of (7.1) near the limit. The cooperative effects in [22] were, however, moreintuited than derived. For example, the typical cooperative term Dε has the form

−Dε[nε](x, t) = −D0 + max(0, D0 −K(x)d(x, {nε ≥ n̄ε)

)(7.4)

where K(x) is a symmetric positive kernel. In that respect, the present work puts the approachin [22] on a more solid framework by actually deriving those effects from realistic biochemicalprocesses.

The present work aims at introducing cooperative terms, similar to those of [22], that arise nat-urally (directly from biological processes), rather than ad hoc mathematical terms. The cooperative

33

effects in the integral operator B[n] in (1.1) appear naturally in the process of model constructionand give a hint of what such terms should look like.

Acknowledgments. P.E. Jabin was partially supported by NSF Grant 1312142, NSF Grant1614537, and by NSF Grant RNMS (Ki-Net) 1107444.

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